Skew products and crossed products by coactions

TL;DR

This paper demonstrates how skew-product graphs and crossed products by coactions relate through universal properties, establishing isomorphisms with tensor products and analyzing amenability of associated actions.

Contribution

It provides a detailed analysis of the isomorphism between skew-product graph C*-algebras and crossed products by coactions, extending results to groupoids and continuous homomorphisms.

Findings

C*(E cross_c G) is isomorphic to the crossed product C*(E) cross_delta G.

The crossed product C*(E cross_c G) cross_gamma G is isomorphic to C*(E) tensor K(l^2(G)).

The action gamma is always amenable under the given conditions.

Abstract

Given a labeling c of the edges of a directed graph E by elements of a discrete group G, one can form a skew-product graph E cross_c G. We show, using the universal properties of the various constructions involved, that there is a coaction delta of G on C*(E) such that C*(E cross_c G) is isomorphic to the crossed product C*(E) cross_delta G. This isomorphism is equivariant for the dual action deltahat and a natural action gamma of G on C*(E cross_c G); following results of Kumjian and Pask, we show that C*(E cross_c G) cross_gamma G is isomorphic to C*(E cross_c G) cross_{gamma,r} G, which in turn is isomorphic to C*(E) tensor K(l^2(G)), and it turns out that the action gamma is always amenable. We also obtain corresponding results for r-discrete groupoids Q and continuous homomorphisms c: Q -> G, provided Q is amenable. Some of these hold under a more general technical condition which obtains whenever Q is amenable or second-countable.